Skip to content# Rearranging Harder Formulae Step-by-Step Examples with Worksheet

## What is Rearranging Harder Formulae?

## Steps for Rearranging the Formula to change the Subject of Formula

## Conclusion

### Practice Questions: Rearranging Harder Formulae

## Solutions:

**Rearranging Harder Formulae**

- Rearranging harder formulae is a valuable skill that empowers us to manipulate equations, isolate variables, and uncover deeper insights into mathematical relationships.
- This skill has practical applications across diverse fields, enabling us to solve complex problems and make informed decisions based on mathematical principles.

In this article, we will discuss:

**What is Rearranging Harder Formulae?****Steps for Rearranging the Formula to Change the Subject of the Formula**

Here is one more link to practice a few extra questions: Maths Genie Rearranging Harder Formulae Questions

- Rearranging Harder Formulae refers to the process of manipulating a given formula to make a specific variable the subject of the formula.
- This process is used extensively in mathematics and science to solve problems where it is necessary to change the variable in a formula to obtain the desired result.

- The following steps can be used to rearrange a formula to change the subject of the formula:

**Step #1:** Identify the variable you want to make the subject of the formula.

**Step #2:** Write down the given formula and identify the variable you want to make the subject.

**Step #3:** Isolate the variable you want to make the subject by performing inverse operations on both sides of the equation.

**Step #4:** Simplify the equation by performing any necessary algebraic manipulations.

**Step #5:** Check your answer by substituting the new expression for the variable back into the original equation to ensure it is correct.

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**Solved Example: **

**Question 1: Rearrange the formula v = u + at to make t the subject.**

Solution:

**Step #1:**Identify the variable you want to make the subject of the formula, which is t.**Step #2:**Write down the given formula and identify the variable you want to make the subject.

**v = u + at**

**Step #3:**Isolate the variable t by performing inverse operations on both sides of the equation.

**v = u + at**

**v – u = at**

**(t = (v – u) / a)**

**Step #4:**Simplify the equation by performing any necessary algebraic manipulations.

**t = (v – u)/a**

**Step #5:**Check your answer by substituting the new expression for the variable back into the original equation to ensure it is correct.

**v = u + a((v – u)/a)**

**v = u + v – u**

**v = v**

The solution is correct.

**Question 2: Rearrange the formula P = (4L + 2W) / 3 to make L the subject.**

Solution:

**Step #1:**Identify the variable you want to make the subject of the formula, which is L.**Step #2:**Write down the given formula and identify the variable you want to make the subject.

**P = (4L + 2W)/3**

**Step #3:**Isolate the variable L by performing inverse operations on both sides of the equation.

**P = (4L + 2W) / 3**

**3P = 4L + 2W**

**(4L = 3P – 2W)**

**(L = (3P – 2W) / 4)**

**Step #4:**Simplify the equation by performing any necessary algebraic manipulations.

**L = (3P – 2W) / 4**

**Step #5:**Check your answer by substituting the new expression for the variable back into the original equation to ensure it is correct.

**P = (4((3P – 2W) / 4) + 2W) / 3**

**P = (3P – 2W + 2W) / 3**

**P = P**

The solution is correct.

**Question 3: Make x the subject of the formula**

Solution:

**Step #1:**Multiply both sides of the equation by (x – b) to eliminate the denominator on the right side.

**a(x – b) = x + c**

**Step #2:**Expand the left side of the equation.

**ax – ab = x + c**

**Step #3:**Move all the terms containing x to the left side of the equation and all the constant terms to the right side.

**ax – x = ab + c**

**Step #4:**Factor out x on the left side of the equation.

**x(a – 1) = ab + c**

**Step #5:**Divide both sides of the equation by (a – 1) to isolate x.

**x = (ab + c) / (a – 1)**

Therefore, x = (ab + c) / (a – 1) is the formula for x in terms of a, b, and c.

- Rearranging harder formulae is a valuable skill that empowers us to manipulate equations, isolate variables, and uncover deeper insights into mathematical relationships.
- By mastering the art of rearranging harder formulae, we develop critical thinking and problem-solving abilities that are essential for success in mathematics and scientific disciplines.
- The ability to rearrange equations allows us to simplify complex expressions, facilitating a clearer understanding of the underlying concepts and principles.
- Through the mastery of rearranging harder formulae, we unlock the potential to tackle challenging mathematical problems, optimize processes, and make accurate predictions based on mathematical relationships.

**Question 1:** Rearrange the formula v = πr^{2}h to solve for h.

**Question 2:** Rearrange the formula A = lw + 2lh + 2wh to solve for w.

**Question 3:** Rearrange the formula T = 2π√(L/g) to solve for L.

**Question 4:** Rearrange the formula E = mc^{2} + mv^{2}/2 to solve for v.

**Question 5:** Rearrange the formula P = 2πh + 2πr to solve for h.

**Question 6:** Rearrange the formula v = πr^{2}h + (4/3)πr^{3} to solve for r.

**Question 7:** Rearrange the formula A = 2πrh + 2πr^{2} to solve for h.

**Question 8:** Rearrange the formula F = G(m_{1}m_{2}/r^{2}) to solve for r.

**Question 9:** Rearrange the formula v = (4/3)πr^{3} – (4/3)πr1^{3} to solve for r.

**Question 10:** Rearrange the formula A = (s/2)(a + b) to solve for b.

Question 1: Rearrange the formula v = πr^{2}h to solve for h.

Solution:

**Step #1: **Start with the formula

**v = πr ^{2}h**

**Step #2: **Divide both sides by πr^{2}

**v / (πr ^{2}) = h**

The rearranged formula is:

**h = v / (πr ^{2})**

Question 2: Rearrange the formula A = lw + 2lh + 2wh to solve for w.

Solution:

**Step #1: **Begin with the formula

**A = lw + 2lh + 2wh**

**Step #2: **Subtract lw and 2lh from both sides.

**A – lw – 2lh = 2wh**

**Step #3:**Factor out w on the right side.

**A – lw – 2lh = w(2h)**

**Step #4: **Divide both sides by (2h).

**(A – lw – 2lh) / (2h) = w**

The rearranged formula is:

**w = (A – lw – 2lh) / (2h)**

Question 3: Rearrange the formula T = 2π√(L/g) to solve for L.

Solution:

**Step #1:**Start with the formula

**T = 2π√(L/g)**

**Step #2:**Divide both sides by 2π.

**T / (2π) = √(L/g)**

**Step #3: **Square both sides of the equation.

**(T / (2π)) ^{2} = L/g**

**Step #4: **Multiply both sides by g.

**g(T / (2π)) ^{2} = L**

The rearranged formula is:

**L = g(T / (2π)) ^{2}**

Question 4: Rearrange the formula E = mc^{2} + mv^{2}/2 to solve for v.

Solution:

**Step #1:** Begin with the formula

**E = mc ^{2} + mv^{2}/2**

**Step #2:**Subtract mc^{2} from both sides

**E – mc ^{2} = mv^{2}/2**

**Step #3:**Multiply both sides by 2/m.

**(2/m)(E – mc ^{2} ) = v^{2}**

**Step #4: **Take the square root of both sides.

**√[(2/m)(E – mc ^{2})] = v**

The rearranged formula is:

**v = √[(2/m)(E – mc ^{2})]**

Question 5: Rearrange the formula P = 2πh + 2πr to solve for h.

Solution:

**Step #1:**Start with the formula

**P = 2πh + 2πr**

**Step #2:**Subtract 2πr from both sides.

**P – 2πr = 2πh**

**Step #3:**Divide both sides by 2π.

**(P – 2πr) / (2π) = h**

The rearranged formula is:

**h = (P – 2πr) / (2π)**

Question 6: Rearrange the formula v = πr^{2}h + (4/3)πr^{3} to solve for r.

Solution:

**Step #1:**Begin with the formula

**v = πr ^{2}h + (4/3)πr^{3}**

**Step #2:**Subtract (4/3)πr^{3} from both sides.

**v – (4/3)πr ^{3} = πr^{2}h**

**Step #3:**Divide both sides by πh.

**(v – (4/3)πr ^{3}) / (πh) = r^{2}**

**Step #4: **Take the square root of both sides.

**√[(V – (4/3)πr ^{3}) / (πh)] = r**

The rearranged formula is:

**r = √[(V – (4/3)πr ^{3}) / (πh)]**

Question 7: Rearrange the formula A = 2πrh + 2πr^{2} to solve for h.

Solution:

**Step #1: **Start with the formula

**A = 2πrh + 2πr ^{2}**

**Step #2: **Subtract 2πr^{2} from both sides.

**A – 2πr ^{2} = 2πrh**

**Step #3: **Divide both sides by 2πr.

**(A – 2πr ^{2}) / (2πr) = h**

**Step #4: **Simplify the expression on the left side.

**A/(2πr) – r = h**

The rearranged formula is:

**h = A/(2πr) – r**

Question 8: Rearrange the formula F = G(m_{1}m_{2}/r^{2}) to solve for r.

Solution:

**Step #1: **Begin with the formula

**F = G(m _{1}m_{2}/r^{2})**

**Step #2: **Multiply both sides by r^{2}.

**Fr ^{2} = G(m_{1}m_{2})**

**Step #3: **Divide both sides by F.

**r ^{2} = G(m_{1}m_{2})/F**

**Step #4: **Take the square root of both sides.

**r = √(G(m _{1}m_{2})/F)**

The rearranged formula is:

**r = √(G(m _{1}m_{2})/F)**

Question 9: Rearrange the formula v = (4/3)πr^{3} – (4/3)πr1^{3} to solve for r.

Solution:

**Step #1: **Start with the formula

**v = (4/3)πr ^{3} – (4/3)πr1^{3}**

**Step #2: **Add (4/3)πr1^{3} to both sides.

**v + (4/3)πr1 ^{3} = (4/3)πr^{3}**

**Step #3: **Multiply both sides by 3/(4π).

**(3/(4π))(v + (4/3)πr1 ^{3}) = r^{3}**

**Step #4: **Take the cube root of both sides.

**r = ∛[(3/(4π))(v + (4/3)πr1 ^{3})]**

The rearranged formula is:

**r = ∛[(3/(4π))(v + (4/3)πr1 ^{3})]**

Question 10: Rearrange the formula A = (s/2)(a + b) to solve for b.

Solution:

**Step #1: **Begin with the formula

**A = (s/2)(a + b)**

**Step #2: **Divide both sides by (s/2).

**A / (s/2) = a + b**

**Step #3: **Simplify the expression on the left side.

**2A / s = a + b**

**Step #4: **Subtract a from both sides.

**(2A / s) – a = b**

The rearranged formula is:

**b = (2A / s) – a**